Paper Info
Title
The hat problem on a directed graph
Abstract
A team of players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his or her own hat color by looking at the hat colors of other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom she or he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph. Previous works focused on the problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique. We show that the conjecture does not hold for directed graphs, and build, for any fixed clique number, a family of directed graphs of asymptotically optimal hat number. We also determine the hat number of tournaments to be one half.
Year
Venue
Keywords
2010
Clinical Orthopaedics and Related Research
clique number.,directed graph,skeleton,hat problem,bipartite graph,complete graph
Field
DocType
Volume
Perfect graph,Discrete mathematics,Block graph,Comparability graph,Combinatorics,Line graph,Clique graph,Forbidden graph characterization,Simplex graph,Directed graph,Mathematics
Journal
abs/1006.1
Citations 
PageRank 
References 
0
0.34
3
Authors
2
Name
Order
Citations
PageRank
Rani Hod1103.28
Marcin Krzywkowski2166.22